Predictable Reversals, Cross-Stock Effects,
and the Limits of Arbitrage
Sandro C. Andrade
U.C. Berkeley
Charles Chang
Cornell University
Mark S. Seasholes
U.C. Berkeley
This Version September 28, 2005∗
Abstract
This paper presents a multi-asset model of stock returns in a market where some investors
place inelastic orders that are uncorrelated with fundamentals. Risk-averse rational arbitrageurs
accommodate trading imbalances but require compensation. The compensation manifests itself
as predictable return reversals. Our model predicts that whenever underlying fundamentals
are correlated, trading imbalances also create cross-stock (temporary) price pressure. We test
implications of the model and show pervasive price pressure at daily, weekly, and monthly
frequencies. A zero-cost portfolio based on sorting trading imbalances into deciles earns a riskadjusted average return of 79 bp per week. Contemporaneous cross-stock trading imbalances
have a price impact that is four times larger for same-industry versus different-industry pairs of
stocks. We end by linking limited risk bearing capacity in the market to approximately 54% of
the observed co-movement in stock returns.
Keywords: Return Predictability, Limits of Arbitrage
JEL number: G12
∗
We thank a national securities firm in Taiwan for providing the margin data; also Brad Barber, Neil Yi-Tsung Lee,
Yu-Jane Liu, and Anlin Chen for providing time-series data of risk factors. We are grateful to Malcolm Baker, Alex
Boulatov, Brad DeLong, Greg Duffee, Darrell Duffie, Robin Greenwood, Terry Hendershott, Ron Kaniel, Arvind
Krishnamurthy, David Levine, Richard Lyons, Darius Miller, and Masahiro Watanabe. Also participants of presentations at: the Emerging Markets Conference at the University of Virginia, Kellogg Finance Department, the
NBER Conference on The Effect of Imperfect Trading on Security Prices, Rice University Finance Department, U.C.
Berkeley Finance Department, and University of Wisconsin Finance Department for helpful discussions. Contact
information: Mark S. Seasholes, U.C. Berkeley–Haas School, 545 Student Services Bldg., Berkeley CA 94720-1900;
Tel: 510-642-3421; Fax: 510-643-1420; email: mss@haas.berkeley.edu.
1
Predictable Reversals, Cross-Stock Effects,
and the Limits of Arbitrage
This Version September 28, 2005
Abstract
This paper presents a multi-asset model of stock returns in a market where some investors
place inelastic orders that are uncorrelated with fundamentals. Risk-averse rational arbitrageurs
accommodate trading imbalances but require compensation. The compensation manifests itself
as predictable return reversals. Our model predicts that whenever underlying fundamentals
are correlated, trading imbalances also create cross-stock (temporary) price pressure. We test
implications of the model and show pervasive price pressure at daily, weekly, and monthly
frequencies. A zero-cost portfolio based on sorting trading imbalances into deciles earns a riskadjusted average return of 79 bp per week. Contemporaneous cross-stock trading imbalances
have a price impact that is four times larger for same-industry versus different-industry pairs of
stocks. We end by linking limited risk bearing capacity in the market to approximately 54% of
the observed co-movement in stock returns.
Keywords: Return Predictability, Limits of Arbitrage
JEL number: G12
1
1
Introduction
Do trading imbalances by one group of investors cause prices to deviate from fundamental
values? Do price deviations exist even if trades are uncorrelated with underlying fundamentals? If so, what are the magnitudes of these deviations and how long does it take prices
to mean-revert back to fundamental values? Finally, does a trading imbalance in one stock
cause prices of other stocks to also deviate away from their fundamental values?
To answer the questions above, we contribute to the limits to arbitrage literature along two
main dimensions. Our first contribution is to provide a multi-asset model that produces
specific testable implications. The model contains a finite number of risk-averse, fullyrational investors who require compensation for accommodating trading imbalances (even
if the imbalances are uncorrelated with asset fundamentals). This compensation manifests
itself as a predictable return reversal. When stocks have correlated fundamentals, trading
imbalances create cross-stock (temporary) price pressure. The cross-stock effect is increasing
in the magnitude of fundamental correlation. Stocks with highly volatile trading imbalances
have higher return volatility after controlling for the variance of fundamentals. A further
implication is that the limited risk-bearing capacity causes excess co-movement of stocks
returns relative to underlying fundamentals.
The second contribution of this paper is to test the implications of our model with a dataset
that is well-suited for testing the empirical implications of limits-to-arbitrage models. Such
models typically assume that rational risk-averse arbitrageurs are able to see (condition on)
the order flow they are absorbing. The arbitrageurs know whether the flow is correlated with
asset fundamentals. Our data allows us to identify a large pool of daily trading imbalances
across hundreds of stocks, over an eight year period. This identification is achieved without
requiring a trade classification algorithm. Most importantly, the data is publicly available
to all investors. And, the source of underlying trades is a specific set of investors that are
unlikely to be trading on the basis of inside information. We describe the data fully in
Section 3 and provide evidence that trades are not linked to underlying fundamentals.
Our analysis provides a clear picture of the economic magnitude of the price pressure and
predictable reversals by means of a sorting methodology. Each week we sort stocks into
deciles based on trading imbalances and form a zero-cost portfolio. Initial price distortions
(as measured by the zero-cost portfolio) are 237 basis points (bp). This is clearly far beyond
transaction cost boundaries. The portfolio exhibits a predictable average return of 72 bp
2
over the week following formation and 188 bp over the five weeks following formation. The
zero-cost portfolio returns cannot be explained by exposure to known sources of systematic
risk and its “alpha” is 79 bp per week on average. While the zero-cost portfolio exhibits high
alpha, we show that portfolio returns are bounded by reasonable after-cost Sharpe ratios,
lending support to the interpretation that risk-averse arbitrageurs do not fully diversify the
exposure created by the absorption of the trading imbalances.
Finally, our paper helps rationalize the high co-movement of stock returns observed in emerging markets. Aggregate risk-bearing capacity is likely to be low in such countries and probably due to a reduced number of arbitrageurs. In our data, correlated fundamentals explains
only 46% of stock return co-movement, while limited risk-bearing capacity is responsible the
remaining 54%. Before introducing our model, we compare and contrast our paper to recent
theoretical and empirical studies.
1.1
Related Theoretical Studies
Our theoretical model can be thought of as a multi-asset extension of Grossman and Miller
(1988) and Holden and Subrahmanyam (2002) that has been stripped down to focus exclusively on how markets absorb trading imbalances. These models have a limited number
of risk-averse arbitrageurs with long horizons. Trading imbalances are positively correlated
with contemporaneous price changes. The same imbalances are negatively correlated with
future price changes. Predictability in stock returns arise from a reduction of the risk premium required for absorbing trading shocks as more information about the stock becomes
available over time. Our multi-asset approach generates the aforementioned price pressure
and reversion. The major differences are the new cross-stock predictions we provide.1
A second strand of models in the limits to arbitrage literature assumes that risk-averse
arbitrageurs behave myopically. These models include DeLong et al. (1990b); Campbell,
Grossman and Wang (1993); Barberis and Shleifer (2003); and Greenwood (2005b). Return
predictability in this framework is generated by the myopia in conjunction with an assumption that trading shocks are not persistent. Myopia may be a reasonable assumption for long
horizons studies due to agency considerations as in Shleifer and Vishny (1997). But such
assumptions may not be appropriate for testing models with high-frequency data (such as
1
Chordia and Subrahmanyam (2004) present a single-asset model similar to the ones just discussed. In their
model, the sign of stock return predictability depends on model parameters due to a second effect. Order imbalances
are (endogenously) positively correlated over time while information continues to become available over time.
3
we have).2 Assumptions about non-persistence do not fit our data as we cannot reject the
hypothesis that trading imbalances are fully persistent.
Greenwood (2005a) develops a model with myopic risk-averse investors in which stock returns follow a predictable pattern after a permanent, uninformed trading shock. The model
assumes arbitrageurs do not consider the possibility of this shock occurring (i.e., expectations
are not fully rational). Such an assumption is reasonable in situations such as the one-time
index re-balancing studied in his paper. But the same assumption is not appropriate for our
paper as we study repeated interactions between liquidity demanders and liquidity providers.
1.2
Related Empirical Studies
Much of the empirical literature surrounding the limits to arbitrage focuses on large rare
events such as additions and deletions from stock indexes—see Harris and Gurel (1986),
Shleifer (1986), Wurgler and Zhuravskaya (2002), and Greenwood (2005a) for some examples.
Recently, papers have turned to identifying liquidity shocks by studying order imbalances
rather than index additions. Chordia and Subrahmanyam (2004) and Chordia, Roll and
Subrahmanyam (2005) use the Lee and Ready (1991) algorithm to identify high-frequency
order imbalances across many stocks and over long time periods.3 The former paper shows
that daily order imbalance is positively correlated with contemporaneous returns for 99%
of NYSE stocks. The authors also find that order imbalances are positively correlated with
one day ahead returns for about 25% of stocks. These results could be due to the fact that
underlying orders may carry information about fundamentals.
We also show pervasive price pressure across the stocks in our market. Our results differ
from the aforementioned papers when it comes to the predictable reversals. An important
difference is that our data are available to all investors each trading day. In our paper, the
source of trading imbalances is a specific set of traders that are unlikely to be trading on
the basis of inside information. In fact, we show empirically that our trading imbalances do
not lead to permanent changes in prices. The empirical results in our paper are also related
to Hasbrouck and Seppi (2001). We extend their work by testing implications of a specific
economic model. One difference is that our model points to lagged trading imbalances (ownstock and cross-stock) as being an important determinant of stock returns.
2
3
We thank Brad DeLong for pointing this out to us.
Lee et. al (2004) also study order imbalances, but use other algorithms to classify trades.
4
The paper now proceeds as follows. Section 2 presents our model and its testable implications. Section 3 describes the data used in this study. Section 4 provides the main results of
the paper regarding trading imbalances and predictable reversals. Section 5 tests additional
cross-sectional implications of the model. It is in this section that we use our trading imbalance data to explain the observed co-movement of stock returns (stock price synchronicity.)
Section 6 shows the results in this paper are robust to numerous alternatives, while the final
sections concludes.
2
Model
Our basic model has two risky assets labeled j and k. Each asset is in unit supply and a
there is a riskless asset in elastic supply. We normalize the interest rate to zero and the price
of the riskless asset to one.
The model has three dates. Agents trade at date 1 and date 2. The risky assets each pay a
liquidating dividend at date 3. For stock j this dividend is equal to S1j + S2j + Φj3 . At date 1,
part of the stock j’s final dividend (S1j ) is revealed to the rational traders. At date 2, another
part of stock j’s final dividend (S2j ) is similarly revealed to the rational traders. The final
part of the dividend (Φj3 ) is revealed at the time of liquidation. Stock k has an analogous
revelation process and dividend structure.
For asset j, the parts of the dividends are: S1j ∼ N (0, Vs ); S2j ∼ N (0, Vs ); and Φj3 ∼ N (0, Vφ ).
For asset k, S1k ∼ N (0, Vs ); S2k ∼ N (0, Vs ); and Φk3 ∼ N (0, Vφ ). The dividends and their
revealed parts are correlated across assets, but are not serially correlated over time:
Corr S1j , S1k
= ρφ
Corr S2j , S2k
j
k
Corr Φ3 , Φ3
Corr S1j , S2j = Corr S1j , Φj3 = Corr S2j , Φj3 = 0
Corr S1k , S2k = Corr S1k , Φk3 = Corr S2k , Φk3 = 0
There are two groups of agents. The first group is referred to as liquidity or noise traders.
They place inelastic orders (demands) for the risky asset at date t ∈ {1, 2}. These orders are
uncorrelated with fundamentals (i.e., uncorrelated with Stj , Stk , Φj , and Φk ). The incremental
demand for asset j at each period of time is denoted by: Ztj ∼ N (0, Vz ) for t ∈ {1, 2}. In
5
other words, the demand shock at date 1 is fully persistent and the total demand at date 2
is Z1j + Z2j for asset j. Similar incremental demand shocks exist for asset k as well. We
assume that the demand shocks are contemporaneously cross-sectionally correlated, but
serially uncorrelated:
Corr Z1j , Z1k
= ρz
Corr Z2j , Z2k
Corr Z1j , Z2j = Corr Z1j , Z2k = Corr Z1k , Z2k = 0
The second group of agents are risk averse investors. We refer to this group as the maximizing
agents, rational investors, or arbitrageurs. These investors maximize date 3 wealth and have
CARA utility with a λ risk aversion coefficient.4
They observe S1j , S1k at date 1 and S2j , S2k at date 2. We denote the demands for risky
assets by the group of risk averse agents by Xtj and Xtk for t ∈ {1, 2}.
2.1
Equilibrium Prices and Returns
Let the prices of the risky assets be Ptj and Ptk for t ∈ {1, 2, 3}. We solve for equilibrium
prices by backward induction (please see Appendix 1 for details). Below, we give expressions
for the prices and return of asset j only.5 Similar expressions exist for asset k as well:
P3j
= S1j + S2j + Φj3
P2j
= S1j + S2j + λVφ (Z1j + Z2j ) + ρφ λVφ (Z1k + Z2k ) − λ (1 + ρφ ) Vφ
P1j
S1j
=
+ λ(Vφ +
Vs )Z1j
(1)
+ ρφ λ(Vφ + Vs )Z1k − λ(1 + ρφ )(Vφ + Vs )
The return to stock j from period 1 to 2 is given by:
P2j − P1j
= S2j + λVφ Z2j + ρφ λVφ Z2k − λVs Z1j − ρφ λVs Z1k + λ (1 + ρφ ) Vs
(2)
Equation (2) contains the main results of the model. We refer almost exclusively to this
equation when formulating testable hypotheses and throughout our empirical section. The
first three terms of the equation represent the surprise part of stock j’s return. The surprise
4
One can think of λ as
λ
,
A
where λ is the absolute risk aversion coefficient of each arbitrageur and A is the total
number of arbitrageurs. The risk-bearing capacity of the market is unlimited if either λ goes to zero or A goes to
infinity.
5
Note that the rational investors know the trading imbalances (Z’s) by conditioning on prices.
6
is determined by three random variables: i) News about fundamentals of stock j; ii) Trading
imbalances to stock j; and iii) Trading imbalances to stock k. The third term drops out if the
two assets have uncorrelated fundamentals (i.e., if ρφ = 0). The last three terms represent
the risk premium earned by the holder of Stock j. Terms four and five represent a conditional
risk premium that arbitrageurs earn for absorbing last period’s trading imbalances. If the
noise traders sold stock j last period, Z1j is negative, and the arbitrageurs earn an extra,
positive return. The last term is the unconditional risk premium associated with stock j.
Note that the unconditional risk premium does not depend on Vz .
Equation (2) highlights the fact that regressions containing returns and own imbalances are
technically mis-specified. Such regressions omit the cross-stock imbalances. This point is
important since many papers focus on a single-asset and own-imbalance effects. Finally, we
envision the length of time between date 2 and date 3 to be much larger than the length of
time between date 1 and date 2 (so that Vφ Vs ).6 Date 3 is far enough into the future so
that the horizon of rational investors coincides with the final resolution of uncertainty about
the risky assets’ payoffs.
2.2
Testable Implications of the Model
Own-Stock Implications: Our model predicts the existence of contemporaneous price
pressure. The risk averse agents require compensation for absorbing the trading imbalances
of the uninformed investors. The financial econometrician observes a positive correlation
between stock j’s returns today and stock j’s trading imbalances today:
Cov P2j − P1j , Z2j = λVφ Vz (1 + ρφ ρz ) > 0
(3)
Even though the trading imbalances are fully persistent from date 1 to date 2, our model predicts the price pressure from the imbalance at date 1 is only temporary. As more information
about the risky asset becomes available, it becomes less risky for the rational arbitrageurs to
absorb a given trading imbalance. The predictable reversal in returns after an initial trading
imbalance can be seen in the following equation:
Cov P2j − P1j , Z1j = −λVs Vz (1 + ρφ ρz ) < 0
(4)
6
This is done for notational simplicity. The same qualitative results would obtain if there were many time periods
of the same length.
7
Cross-Stock Implications: Our model predicts that trading imbalances to stock k affect
the returns of stock j if the two stocks have correlated dividends (fundamentals). The
intuition behind this result comes from thinking about the rational arbitrageurs’ hedging
possibilities. Suppose the noise traders decide to sell stock k. Risk averse and competitive
arbitrageurs agree to accommodate the imbalance, but require compensation. The price of
stock k is bid down. However, if fundamentals are correlated, arbitrageurs may partially
offset the risk of absorbing shares of k by selling some of stock j. This mechanism transfers
part of the trading imbalance from stock k to stock j. When fundamentals are positively
correlated, a negative trading imbalance to stock k creates a negative price pressure in
stock j.
Actual Markets and Multiple Assets: In Section 4 we test our model with stock
market data. Actual markets have more than two stocks so we expand the basic model
from two to N risky assets, under the simplifying assumption that all stocks share the same
pairwise correlation of fundamentals (ρφ ). The multi-asset extension of our model extends
Equation (2) from above to:
P2j − P1j
= S2j + λVφ Z2j + ρφ λVφ
X
Z2k − λVs Z1j − ρφ λVs
k6=j
X
Z1k + const.
k6=j
The equation directly above implies that we want to regress returns on current and lagged
trading imbalances. As Section 3 describes, we have a balanced panel with 131 companies
and 443 weeks of data. Estimation of the equation directly above requires 262 right hand
P
side variables which is not desirable. Notice that if ρZ = 0 then k6=j Ztk → 0 for N → ∞
by the Law of Large Numbers. Assuming ρZ 6= 0 the cross imbalances do not tend to cancel
each other out as we consider more and more stocks. If there is a factor structure to the
cross imbalances (the Z2k ’s) then we can reduce the number of right hand side variables.
In the next section, we show that ρz > 0 on average and there is a single, dominant factor to
the trading imbalances. Stocks loadP
evenly on this factor which allows us to substitute the
∗
EW ∗
following expression (N −1)Zt
≈ k6=j Ztk . Here, ZtEW denotes the time t equal-weighted
average imbalance across all stocks in the market except stock j.
P2j − P1j
∗
∗
= S2j + λVφ Z2j + ρφ λVφ (N − 1)Z2EW − λVs Z1j − ρφ λVs (N − 1)Z1EW + const.
In reduced form, the above equation becomes our main estimation and is shown below. We
can use this reduced form estimation equation to test the impact on returns of own-stock
imbalances and cross-stock imbalances:
∗
∗
j
EW
rtj = π0 + π11 Ztj + π12 (N − 1)ZtEW + π21 Zt−1
+ π22 (N − 1)Zt−1
+ εjt
8
(5)
3
Data
We obtain data with the help of national securities firm in Taiwan and from a data provider
called the Taiwan Economic Journal (TEJ). All price and trading data are from the Taiwan
Stock Exchange (TSE). The date are available on a daily basis from January 1, 1994 to
August 29, 2002, accounting for 2,360 trading days or 443 calendar weeks. The data cover
all 647 listed companies in Taiwan. In order to account for occasional trading suspensions
of a few stocks, we consider both the complete dataset as well as a sub-sample of the 131
stocks. The sub-sample has price and trading data available throughout the entire sample
period and thus provides a balanced panel.7
Price and Return Data: We obtain share prices, returns, and number of shares outstanding for each listed stock at a daily frequency. Data are available from the TEJ. Daily and
weekly returns are adjusted for capital changes.
Trading Data: We obtain the net number of shares bought or sold by margins traders on
a daily basis for 608 stocks.8 The number of shares are reported to the TSE by brokerage
firms. The aggregate number of shares bought or sold (net) for each listed stock is available
each day from the TSE on the exchange’s website and from other data providers such as the
TEJ.
We use the net number of shares bought (sold) for stock j as the empirical analog to Ztj
from our model. In order to compare quantities across firms, we typically normalize by the
number of shares outstanding.
Ztj ≡
Net Shares Bought(Sold) jt
Total Shares Outstanding jt
To control for outliers in regressions, we winsorize the above measure at the 0.50% and
99.50% levels for the daily and weekly data. The change is irrelevant with non-parametric
sorting tests. The aggregate holdings of stock j on date t is the sum of all past net trading.
P
We have the holdings each day and week for each stock in our sample: Htj ≡ tτ =−∞ Zτj .
7
Results remain qualitatively similar when using both the full sample and sub-sample and Section 6.9 provides
specific robustness tests. Appendix 2 provides additional description about the sub-sample selection procedure.
8
There are 39 stocks that have no margin trading.
9
3.1
An (Ex-Ante) Empirical Analog of Ztj
There are many reasons why our net trading data are an appropriate empirical analog of Ztj .
First, it is estimated that 99.3% of our margin trading data come from individuals rather
than from institutions.9 Several papers in the literature advance the idea that individuals
trade for reasons unrelated with stock fundamentals—see Barber and Odean (2000). Using
data from 1994 to 1999, Barber et. al. (2005) present compelling evidence that individuals
in Taiwan trade too much and on average lose money in their trading. Annual turnover
is 300% to 600%, and reduces returns by 3.5% per year. The authors point out that it is
unlikely that private information, re-balancing or hedging needs of individuals can account
for such numbers.
We build a representative portfolio based on the weekly long margin holdings data (Htj ).
This portfolio under-performs the market by 10.67 bp per week, adding up to a loss of
over 5% per annum.10 The turnover of the representative portfolio is approximately 250%
per year. Most importantly, Barber et. al. (2005) have micro-structure data from Taiwan
and classify all trades as either “aggressive” or “passive”. Barber graciously calculated some
summary statistics for net margin trading in the market. He reports that approximately 90%
of the individual trades underlying our daily data can be classified. Of these, approximately
80% can be classified as “aggressive”. These statistics show that the trades are demanding
liquidity (not supplying it).
The rational investors of our model know the trading imbalance. Such assumption is quite
common in limits-to-arbitrage models. Our margin trading data is disclosed to investors,
rather than being proprietary to the TSE. Investors can see the full margin trading imbalance
each day.
Finally, our statistical results would have low power if margin trading was responsible for
just a tiny fraction of the Taiwanese stock market. Fortunately this is not the case. Our
margin trading data account for a large fraction of the trading volume in the TSE. Within
9
It is important to note that this is not a paper about the average performance of all individual investors. Instead,
this examines a well-defined subset of investors for whom we can identify trading imbalances. It happens that the
subset is populated 99.3% by individuals which, in part, gives us confidence that the data maps rather cleanly into Ztj .
The full set of reasons for believing this is a clean mapping are listed in this subsection, Section 4.7, and Appendix 3.
10
The under-performance has a -2.08 T-statistic. On the one hand, the fact the T-statistic is not too big is good
otherwise one might worry trades are motivated by (bad) information. On the other hand, Section 4.1 of this paper
shows the price impact “paid” by the margins traders can be substantial. It is this price impact which undoubtedly
contributes to the under-performance.
10
the sub-sample of 131 companies, margin trading accounts for an average of 42.36% of a given
stock’s turnover. Appendix 3 provides further additional tests that lend ex-ante support to
the use of margin trading imbalances as the empirical analog of Ztj in our model. We show
that net margin trades are uncorrelated with changes in the six month money market rates,
commercial paper rates, loans at domestic banks, and the change in the premium on the
Taiwan (equity) fund. Of course, we test the predictions of the model explicitly in Sections 4,
5, and 6. Confirmation of the model’s predictions can be thought of as ex-post support for
using the margins trading data as an empirical analog of Ztj . We revisit the issue of ex-post
evaluation in Section 4.7 after first carrying out the tests.
3.2
Overview Statistics
P j
Table 1, Panel A shows cross-sectional measures of Ztj , ∆Ztj , and Htj =
Zt . We see
that the net trading is, on average, near zero. But, net trading is clearly volatile as ∆Ztj is 0.0013 per day on average and 0.0039 per week on average. On average, investors in our
sample hold 0.0725 of shares outstanding, with a 0.0360 to 0.1049 intra-quartile range. There
is not much difference between cross-sectional measures if we consider the full sample of 608
firms or if we consider the sub-sample of 131 firms.
Table 1, Panel B shows the time-series means and standard deviations of an equally-weighted
trading measure (across stocks.) The standard deviation of net trading is analogous to Vz
from our model. The unbalanced panel accounts for the disparity in average holding levels
between Panel A and Panel B for the full sample of firms. For the balanced sub-sample of
131 companies, sample averages are equal across panels.
Table 1, Panel C shows a break-down of the firms in our sample by industry. In Sections 5.1
and 5.2 we use industry classification as a proxy for correlated fundamentals. Taiwan has
seen a large growth in its electronic industry, many of which are first listed late in our
sample period. The balanced sub-sample of 131 firms represents over 40% of total market
capitalization at the end of 1998.
Table 2 shows results that provide further justification for using our net trading data as the
empirical analog to Ztj from our model. The model assumes trading imbalances Z1j , Z2j are
fully persistent. To test this assumption, we perform a unit root test of the null hypothesis
P j
that the level of holdings, Htj =
Zτ , follows a random walk. Using daily data and our
balanced panel, Panel A shows we can only reject the null for 3 of the 131 stocks (at the
11
5%-level). Using weekly data, we can only reject the null for 7 of the 131 stocks (at the
5%-level). Clearly these rejection rates are what one would expect by pure chance.
In Section 2.2 we discuss the testable implications of our model in an actual stock market
with many assets. One requirement for cross-stock trading imbalances to matter is that
ρz is different from zero (i.e., trading imbalances are contemporaneously cross-sectionally
correlated.) We test this in Table 2, Panel B. The average pairwise correlation is 0.0612
using daily data and 0.1209 using weekly data. Our sub-sample has 131 stocks which gives
8,515 pairs. Of the 8,515 pairs, only 291 pairs (or 3.42%) have negatively correlated trading
imbalances at a daily frequency and only 343 pairs (or 4.03%) have negatively correlated
trading imbalances at a weekly frequency. The average pairwise correlation is significantly
different from zero at all conventional levels.
In Table 2, Panel C we show the results of a principal component analysis. We report the
fraction of explained contemporaneous correlation of trading imbalances (Ztj ). The first principal component is clearly dominant and additional principal components are economically
∗
negligible (especially at the weekly frequency). This finding supports our use of ZtEW in
Equation 5—our main regression equation. In other words, trading imbalances do not offset
each other even when the number of stocks gets large.11 We test for statistical significance
using the asymptotic approximation outlined in Hasbrouck and Seppi (2001). At a daily
frequency, the first eigenvalue is 9.5181 (and the first principal component explains 7.27%
of observed correlation where 0.0727 = 9.5181 ÷ 131.) We can approximate the standard
p
error with 2(9.5181)2 /2359 = 0.28 which corresponds to a 34.34 Z-statistic. At a weekly
frequency, the first eigenvalue is 17.6487 (and the first principal component explains 13.47%
of correlation where 0.1347 = 17.6487 ÷ 131.) We can approximate the standard error with
p
2(17.6487)2 /443 = 1.19 which corresponds to a 14.88 Z-statistic.
4
Trading Imbalances and Predictable Reversals
We use our main regression equation developed in Section 2.2 and shown in Equation (5) to
test the implications of our model. Although we use a single regression framework we present
results in four sub-sections below: 4.1) Own-stock contemporaneous effects; 4.2) Own-stock
reversals; 4.3) Cross-stock contemporaneous effects; and 4.4) Cross-stock reversals. We
consider both daily and weekly frequencies. For most of this section, we consider N =
11
Results not reported, but available upon request, show that stocks load evenly on the first principal component.
12
131 stocks in our sub-sample. Results using the full sample of 608 stocks are checked in
Section 6.9. All T-statistics reported in the four sub-sections below are based on standard
errors that allow for heteroscedasticity and clustering of contemporaneous observations (i.e.,
across stocks, at the same point in time.)
4.1
Own-Stock, Contemporaneous Price Impact
We test if signed trading imbalances of stock j are positively correlated with contemporaneous returns of stock j. This test is a direct implication of our model and can be seen in
Equation (2) and Equation (3). The simplest test involves a pooled regression of returns
rtj on a constant and own trading imbalances Ztj . In Table 3, Regression 1 we see the
own trading imbalances have a coefficient (π11 ) of 2.7014 with a 44.65 T-statistic when using
daily data. In Regression 4, the own trading imbalances have a coefficient of 2.6843 with a
17.83 T-statistic when using weekly data. As predicted, we find a strong positive relationship
between a stock’s own trading imbalances and it’s contemporaneous returns.12
We estimate the correctly specified form of Equation (5) in Regression 3 (daily) and Regression 6 (weekly). The own-stock coefficients (π11 ) are now 2.0017 with a 54.39 T-statistic
(daily) and 1.5493 with a 20.33 T-statistic (weekly). The adjusted-R2 of our regression is
0.1730 at a daily frequency and over 0.2128 at a weekly frequency. To initially quantify the
economic magnitude of these results, we note that a one standard deviation increase to this
week’s Ztj leads to a 91 bp increase in this week’s stock price. We explore the economic
magnitude more fully in Section 4.5 below.
4.2
Own-Stock, Predictable Reversals
We test if trading imbalances of stock j are negatively correlated with future returns of
stock j. This test follows directly from our model and can be seen in Equation (4). Again,
the simplest test involves a pooled regression of current returns rtj on a constant and lagged
j
own trading imbalances Zt−1
. In Table 3, Regression 2 we see stock j’s own-stock lagged
trading imbalance has a coefficient (π21 ) of -0.4684 with a -8.65 T-statistic when using daily
12
As robustness checks, we re-calculate all standard errors in Table 3 allowing for clustering by stocks. We also
re-estimate all coefficients and standard errors based on firm-fixed effects and clustering of contemporaneous observations. Finally Table 3, Regressions 1, 2, 4, and 5 are re-estimated using Fama-McBeth methodology. In all cases,
results are not materially different from those shown in Table 3.
13
data. In Regression 4, the own-stock lagged trading imbalance has a coefficient of -0.4250
with a -3.13 T-statistic when using weekly data. These own-stock results give the first
indication of predictable reversals and a clear pattern emerges. Positive changes in Ztj are
followed by negative returns next period. Negative changes in Ztj are followed by positive
returns.
We estimate the correctly specified form of Equation (5) in Regression 3 (daily) and Regression 6 (weekly). The own-stock lagged coefficients (π21 ) are now -0.7972 with a -26.65
T-statistic (daily) and -0.4709 with a -7.82 T-statistic (weekly). Notice that the magnitudes
of the coefficients on own-stock lagged imbalances are about 65% less than the magnitudes
of the coefficients on own-stock contemporaneous imbalances (0.7972 vs. 2.0017 at a daily
frequency and 0.4709 vs. 1.5493 at a weekly frequency). To better understand the economic magnitude of the predictable reversals we use a non-parametric sorting methodology
in Section 4.5 below.
4.3
Cross-Stock, Contemporaneous Price Impacts
We test if trading imbalances in stock k impact the returns of stock j as predicted by our
model. We use the full form of Equation (5) to estimate both own-stock and cross-stock
price impacts, but concentrate on the contemporaneous cross-stock term (π12 ).
The results in Table 3, Regression 3 (daily) and Regression 6 (weekly) provide strong support
for our model. At a daily frequency, the estimated contemporaneous cross-stock parameter
(π12 ) is 0.1542 with a 21.54 T-statistic. At a weekly frequency, the estimated contemporaneous cross-stock parameter (π12 ) is 0.1152 with a 16.08 T-statistic. Thus, a positive
imbalances in stock k (where k 6= j) increases stock j’s returns while a negative imbalances
in k decreases stock j’s returns. To initially quantify the economic magnitude of these re∗
sults, we note that a one standard deviation imbalance to (N − 1)ZtEW leads to a 283 bp
price movement in stock j using weekly data. This value is over three-times larger than the
91 bp own-stock impact reported in Section 4.1 above.
4.4
Cross-Stock, Predictable Reversals
Our model predicts that cross-stock effects also play a role in the predictable pattern of
return reversals. Trading in stock k today impacts the returns of stock j today which
14
partially adds to stock j’s mean-reversion in the future. Table 3 Regression 3 (daily) and
Regression 6 (weekly) show that cross-stock effects lead to predictable reversals. The lagged
cross-stock parameter (π22 ) is -0.0562 with a -7.84 T-statistic at a daily frequency. The lagged
cross-stock parameter (π22 ) is -0.0536 with a -7.88 T-statistic at a weekly frequency. Again,
the magnitude of the lagged cross-stock parameters are much smaller than the magnitude
of the contemporaneous cross-stock parameters (0.0562 < 0.1542 daily; 0.0536 < 0.1152
weekly). We now turn to a sorting procedure to quantify the economic magnitude of the
return predictability.
4.5
Sort Results and Predictable Return Reversals
In order to quantify the economic magnitude of price distortions and mean-reversion, we
perform a simple sorting procedure based on trading imbalances. The sort procedure also
controls for outliers in Ztj . Sorting is a non-parametric procedure that allows us to detect
non-linear price pressure. Finally, the sort procedure allows us to account for cross-stock
effects without having to specify them in a regression framework.
Each period we rank stocks based on Ztj . Stocks are then put into one of ten portfolios
based on ranking. The lowest decile is called “Portfolio 1” and Ztj of stocks in this portfolio
usually have a negative sign, though not necessarily. Likewise, the highest decile is called
“Portfolio 10” and Ztj of stocks in this portfolio usually have a positive sign, though also not
necessarily. Our procedure is similar to the typical sorting methodology used in momentum
studies.
In Table 4, we measure the equal-weighted return to each of the ten portfolios (deciles) over
the following period for the sub-sample of 131 stocks. We report returns over the next day
and over the next week. Stocks with low Ztj this period have high returns next period; stocks
with high Ztj this period have low returns next period. For example, stocks in the lowest
weekly decile go up 39 bp on average over the following week. Stocks in the highest weekly
decile go down 33 bp over the following week.
Table 4 also reports the results for a hypothetical zero-cost portfolio that goes long low
Ztj stocks and short high Ztj stocks. On average, the zero-cost portfolio returns—labeled
“Difference 1-10” or r1−10 —are 42 bp per day and 72 bp per week. T-statistics are based
on Newey-West standard errors and indicate a high level of significance at both frequencies.
Our full sample has 443 weeks of holdings data and 442 weeks of trading data (changes to
15
holdings). The sorting procedure uses one week which leaves 441 weeks of returns as shown
in the table.
While Table 4 only shows next-period returns, we track the zero-cost portfolio returns over
the next ten periods. Figure 1 shows the cumulative returns to the zero-cost portfolio
(inverted) for the sub-sample of 131 firms. We invert in order to aid intuition. Think of the
graph as follows: at time zero, uninformed traders heavily buy (sell) certain stocks. The
prices shoot up (down) a combined total of 129 bp that day or 237 bp that week. Prices
then start to mean-revert back to pre-existing levels. On the first day following portfolio
formation, prices fall (rise) 42 bp. At a weekly frequency, prices fall (rise) 72 bp during the
first week following portfolio formation. The 42 bp and 72 bp are exactly the returns to the
zero-cost portfolio shown in Table 4. The mean-reversion doesn’t stop after only one period.
Over the five to six weeks that follow the initial buying (selling), prices continue to fall (rise)
toward pre-existing levels.13
4.6
Risk-Adjusted Sort Returns
We examine the relationship of the zero-cost portfolio with known and potential risk factors.
In Table 5, we regress the excess weekly returns of the zero-cost portfolio on a constant and
the market excess return. Regression 1 shows the alpha (α) of the regression is 73 bp which
is very similar to the 72 bp reported in Table 4. The market beta is low and not significantly
different from zero.
In Table 5 we also include Fama-French size (SMB) and book-to-market (HML) factors as
well as a Carhart momentum factor (MOM).14 The table shows that regression coefficients
relating to possible factors are insignificant, except for beta on the momentum factor. We
13
Figure 1 makes it easy to see how little evidence there is of private information being incorporated (on average)
into stock prices. As opposed to models of insider information such as Kyle (1985), prices mean-revert almost
completely back to pre-existing levels. The 237 bp price impact in week zero comes from both contemporaneous
own-stock effects and contemporaneous cross-stock effect. Since trading imbalances are not perfectly correlated, the
economic magnitudes discussed in Section 4.1 and Section 4.3 cannot simply be added together. The one standard
deviation imbalance mentioned in these earlier sections is not the same as being included in the zero-cost portfolio.
14
Appendix 5 provides a a description of factor construction. We note, however, that Chen and Zhang (1998) look
at Taiwanese stock returns between 1976 and 1993 and conclude that neither size-sorted nor book-to-market-sorted
zero-cost portfolios earn statistically significant profits. This is confirmed in Barber, Lee, Liu and Odean (2004),
who show that profits of size, book-to-market, and momentum-sorted portfolios change sign in their 1983-2002 and
1995-1999 sub-samples. Therefore, the evidence that SMB, HML and MOM are proxies for sources of systematic risk
is weak in Taiwan.
16
conclude that any serial correlation that exists in the market does not affect the risk-adjusted
alpha of the zero-cost portfolio. As can be seen in Table 5, Regression 5 the alpha is 79 bp.
Thus, the fully risk-adjusted value is actually greater than the non risk-adjusted alpha of
72 bp as shown in Table 4. Additional tests, discussed in Section 6.3 and Section 6.4, show
that the mean reversion is not the product of momentum, feedback trading, or volume-based
strategies. In Section 5.4 and Section 6.5, we show that returns cover transaction costs after
an initial one-week holding period.
4.7
An (Ex-Post) Empirical Analog of Ztj
Section 2 models stock returns when liquidity suppliers have limited risk bearing capacity.
They accommodate order flow that is uncorrelated with stock fundamentals. The liquidity
suppliers in our model know that such order flows do not carry information about stock
fundamentals—a common assumption in limits-to-arbitrage models. We then test implications of the model with net trading imbalances of margin traders.
Section 3.1 and Appendix 3 give ex-ante evidence that our trading imbalances are uncorrelated with fundamentals. The tests in this section provide ex-post evidence that our trading
imbalances are uncorrelated with stock fundamentals. The imbalances are associated with
statistically significant and economically large price pressure at both daily and weekly frequencies. Prices tend to revert back to pre-existing levels in a predictable manner over a
five week horizon. If the imbalances were correlated with a factor in the economy, the factor
would need to exhibit mean-reversion at daily, weekly, and monthly frequencies. We find
this unlikely and now turn to testing further implications of our model.
5
Additional Empirical Implications of Our Model
Our modeling framework produces three additional, cross-sectional implications which we
test below: 5.1) A stock’s return variance is shown to be increasing in the variance of it’s
own trading imbalances; 5.2) The magnitude of the of the cross-stock impact is shown to
be increasing in the correlation of fundamentals; and 5.3) The observed co-movement of
stock returns can be explained, in part, by trading imbalances. We end this section with an
analysis of the limits to arbitrage and after-cost Sharpe ratios.
17
5.1
Own-Stock Variance
We test that a stock’s return variance is increasing in the variance of it’s own trading imbalances. To test this, we relax an earlier assumption of our basic model and allow different
variances of trading imbalances across stocks (e.g., Vzj and Vzk ). Equation (2) remains unchanged. However, the variance of stock j’s return becomes:
1
1
V ar(P2j − P1j ) = VS + λ2 (Vφ2 + Vs2 ) Vzj + ρ2φ Vzk + 2ρz ρφ (Vzj ) 2 (Vzk ) 2
Note that when ρφ and ρz are small, the variances of two different stocks in the market
become:
V ar(P2j − P1j ) ≈ VS + λ2 (Vφ2 + Vs2 )Vzj
V ar(P2k − P1k ) ≈ VS + λ2 (Vφ2 + Vs2 )Vzk
The equations above show that, after controlling for variance of fundamentals, stocks subjected to more volatile trading imbalances have higher return volatility. We control for
fundamentals by comparing stocks within the same industry. We use the TSE industry
classification system, which maps our 131 stocks into nineteen different industries. We
first calculate the variance of returns—V ar(rtj )—and the variance of trading imbalances—
V ar(Ztj )—for each stock. We then compute the standardized variance of returns and the
standardized variance of trading imbalances within each industry.15
The Spearman rank correlation of the standardized variance of returns and the standardized
variance of trading imbalances for all the 131 stocks is 0.5880 with a 0.0000 p-value. Note the
Spearman rank correlation of unstandardized variances is 0.6864 with a 0.0000 p-value. Our
results support the model’s prediction the variance of returns increases with the variance
of own trading imbalances. The results remain unchanged after controlling for different
variance of fundamentals using industry classifications.
5.2
Magnitude of Cross-Stock Impact
We test if the magnitude of cross-stock impacts is increasing in the correlation of fundamentals. To do this, we extend our multi-asset model with N stocks by allowing different
pairwise correlations of fundamentals across stocks (ρj,k
φ ). In this extension, Equation (2)
15
X−cross-sectional average
To standardize a firm’s measure “X” we take ( cross-sectional
).
standard deviation
18
becomes:
P2j − P1j
= S2j + λVφ Z2j + λVφ
X
j
k
ρj,k
φ Z2 − λVs Z1 − λVs
k6=j
X
k
ρj,k
φ Z1 + const.
k6=j
If we focus on a single pair of stocks (j and k), we can re-write the equation above as:
P2j − P1j
k
= S2j + λVφ Z2j + λVφ ρj,k
φ Z2 + λVφ
X
j,k k
j
i
ρj,i
φ Z2 − λVs Z1 − λVs ρφ Z1 − λVs
i6=j,k
X
i
ρj,i
φ Z1 + const.
i6=j,k
This second equation makes it clear that the magnitude of the cross-stock impact is increasing in the fundamental correlation (ρj,k
φ ). In other words, trading imbalances to k have
an increasingly large impact on stock j’s returns as ρj,k
φ increases. Relying once more on
the existence of a single factor structure in trading imbalances, we can make the following
P
∗∗
substitution (N − 2)ZtEW ≡ i6=j,k Zti and write:
rtj = π0 + π111 Ztj + π112 Ztk + π113 (N − 2)ZtEW
∗∗
j
k
EW
+ π121 Zt−1
+ π122 Zt−1
+ π123 (N − 2)Zt−1
∗∗
+ εjt
rtk = π0 + π211 Ztj + π212 Ztk + π213 (N − 2)ZtEW
∗∗
j
k
EW
+ π221 Zt−1
+ π222 Zt−1
+ π223 (N − 2)Zt−1
∗∗
+ εkt
We form all 8,515 pairwise combinations with the 131 stocks in our sub-sample and estimate
the set of equations above for each pair. For a given pair of stocks j and k, we define the
211
contemporaneous cross-stock coefficient as: CCC j,k ≡ π112 +π
. The model predicts that
2
j,k
CCC is increasing in the correlation of fundamentals.
We test the prediction under the assumption that two stocks in the same industry have more
highly correlated fundamentals than two stocks in different industries. As Table 6 shows,
the average CCC j,k is 0.2851 across all 8,515 pairs. The average CCC j,k is 0.9973 across the
567 pairs for which j and k are from the same industry. The average CCC j,k is only 0.2342
across the 7,948 pairs for which j and k are not from the same industry. We also report
the median value of CCC j,k and the standard deviation of CCC j,k across pairs. It is clear
that we do not have 8,515 independent observations. However, even if we assume that we
have only 131 independent observations (pairs), we can reject that CCC j,k is the same for
same-industry and different-industry pairs at the 5%-level.16 Our results support the model’s
prediction the magnitude of cross-stock impact increases with fundamental correlation. Our
estimate of the cross-stock impact (CCC j,k ) is over four times larger when stocks are from
the same industry than when they are not.
16
Assume the distribution of same-industry/different-industry pairs is the same as is shown as the bottom of Table 6.
In other words, assume there are nine independent pairs with j and k in the same industry and 122 independent
pairs with j and k in different industries.
19
5.3
Co-Movement
We show that our model and trading imbalance data can help explain the observed comovement of stock returns. Recent work by Morck, Yeung, and Yu (2000) has focused
attention on this topic—especially in emerging markets. While we cannot do a cross-country
comparison due to the lack of similar trading data from other countries, we can test what
fraction, if any, of observed co-movement is explained by our trading imbalances. To do this,
we begin by measuring the R2 on a stock-by-stock basis. The following equation regresses
an individual stock’s excess returns on a constant and the market’s excess returns:
rtj − rtf = α + β rtm − rtf + εjt
(6)
Table 7, Specification 1 shows the average R2 is 0.2958 using weekly data from the 131 firms
in our sub-sample. We next augment Equation (6) by orthogonalizing the market excess
return (rtm − rtf ) with respect to our market-wide measure of trading imbalances ZtEW .
EW
The new measure is labeled rtm⊥Z . For each stock j, we also decompose ZtEW into a
portion that is parallel to Ztj and a portion that is orthogonal Ztj . We refer to these two new
EW kj
measures as Zt
and ZtEW ⊥j respectively. The new regression equation is thus:
rtj − rtf = γ0 + γ1 rtm⊥Z
EW
EW kj
+ γ2 Zt
+ γ3 ZtEW ⊥j + νtj
(7)
Table 7, Specification 2 shows that the average R2 is only 0.1630 when using the orthogonalized market return on the right-hand side. Specification 5 produces an average R2 0.3474
when all three of our new measures are used. We view the co-movement results in Table 7 as
another way to present implications of our model. Equation (2) shows that stocks co-move
for three reasons: 1) Fundamental information is correlated across stocks; 2) Own trading
imbalances are correlated across stocks; or 3) There are cross-stock effects which induce
excess co-movement above the previous two effects. From the reported R2 measures in Table 7, we conclude that market returns (correlated fundamentals) only explain about 46%
of observed co-movement. Correlated trading imbalances explain approximately 23.75% of
observed co-movement. Cross-stock effects explain the remaining 30.25% of observed comovement.17
Both the own trading imbalances and the cross-stock effects are important factors whenever
a market has aggregate risk-bearing capacity (i.e., limits to arbitrage.) If we were to have
17
Our decomposition of observed co-movement is based on the following (rough) approximations. The approxima-
tions can define either denom = 0.3474 or denom = 0.1630 + 0.0827 + 0.1062 = 0.3519. The observed co-movement
can then be explained by:
0.1630
denom
' 0.46;
0.0827
denom
' 0.2375; and
20
0.1062
denom
' 0.3025
similar data from other countries, we could try to link the strength of the trading imbalances
to country-level variables such as investor protection or prevalence of institutional investors.
Even without data from other countries, our model provides a unified framework for analyzing co-movement. This framework points to aggregate risk-bearing capacity as being a key
factor.
5.4
After-Cost Sharpe Ratios
In our model prices deviate from fundamental values. The fundamental value of stock j at
date 2 is S1j + S2j while the price, as seen in Equation 1, is also a function of λ, Vφ , Ztj , Ztk ,
Vs , and ρφ . This deviation is driven by aggregate risk-aversion in the market. Equation 1
shows that deviations disappear if there is unlimited risk-bearing capacity in the market (in
other words, if λ = 0).
We investigate the effect of risk-aversion further. Figure 2, Panel A plots the cumulative
return to the zero-cost portfolio based on our sort results. The first week’s return is 72 bp
as shown in Table 4. Figure 2 shows that holding the zero-cost portfolio for two weeks or
longer covers transaction costs.18
If prices are driven too far from fundamentals, we expect optimizing investors to step-in until
the expected Sharpe ratio of the zero-cost (risky) portfolio is reasonable. In Panel B, we
include the iso-bars of four different after-cost Sharpe ratios (plotted on the same time vs.
pre-cost return axes we use in the left-hand panel.) The steepest line shows the return needed
to achieve an annual after-cost Sharpe ratio of 0.50 and holding the zero-cost portfolio for one
week, two weeks, three weeks, etc. The returns of the zero-cost portfolio are bounded above
by an after-cost Sharpe ratio of 0.50 (approximately.)19 The fact that after-cost Sharpe ratios
are bounded by economically reasonable values provides indirect support that risk-bearing
capacity is an important limit to arbitrage.
18
19
Section 6.5 explains how we calculate transaction costs.
To achieve an after-cost Sharpe ratio of 0.50 after eight weeks, the zero-cost portfolio would need to earn 239 bp,
as indicated by the “” in the Figure 2, Panel B. Holding the zero-cost portfolio for one week does not cover costs,
thus the after-cost Sharpe ratios are negative and not shown.
21
6
6.1
Robustness Checks
Ratio of Coefficients
The return equation for Stock j—Equation (2)—and our main estimation regression—
Equation (5)—imply estimated coefficients should be in a fixed ratio:
π12
ρφ λVφ
π22
ρφ λVs
=
=
=
= ρφ
(8)
π11
λVφ
π21
λVs
We test this restriction when estimating Regression 3 and Regression 6 in Table 3. When
using daily data, we fail to reject the restriction at the 5%-level (the p-value is 0.3503) which
provides additional support for the model. Using weekly data we are able to reject at the 5%
level. Despite this rejection, we note that the ratio of weekly coefficients shown in Table 3,
Regression 6 is economically close: 0.1152
≈ −0.0536
.
1.5493
−0.4709
6.2
Expanded Sort Results
Appendix 4 contains expanded sort results. We sort stocks into ten deciles based on the past
1, 2, or 3 three values of Ztj at both daily and weekly frequencies. We form a portfolio that
goes long the bottom decile and short the top decile of the sorted stocks. We then track that
portfolio for the next 1, 2, or 3 periods (days or weeks).
If we concentrate on the weekly frequency, we see the zero-cost portfolio earns 72 bp if the
sort is based on the past one week and we track returns for one week in the future (the same
as in Table 4). If we sort by the past three weeks and track returns for three weeks into the
future, the return goes up to 171 bp.
Finally, the daily and weekly results extend to a calendar monthly frequency (results are not
shown in Appendix 4 but are available upon request). If we sort stocks by the past month’s
Ztj , the zero-cost portfolio earns an average of 154 bp over the next month. Like earlier
results, this average return is statistically significant and has a 2.54 T-statistic.
6.3
Momentum in Returns
Does the zero-cost portfolio unintentionally capture momentum (or reversals) in stock returns? No. Table 5 addresses this question directly. We can see that the zero cost portfolio
22
loads negatively on the momentum factor, while the statistical and economic significance of
the alpha in the regression remains virtually unchanged.
We investigate further. It turns out that Taiwanese stock returns are, on average, slightly
positively auto-correlated at a daily frequency and Appendix 6, Panel A shows this clearly.
We believe this effect is driven by small, low-priced stocks and the daily price limits that exist
in this market. Our sorting procedure produces mean reversion at a daily frequency despite
the positive auto-correlation. At a weekly frequency, there is no significant auto-correlation
in returns. We simply cannot use weekly returns to forecast future mean-reversion. As
Panel A shows, at most, 15 bp of the 72 bp (mean-reversion) reported in Table 4 can be
attributed to underlying patterns in returns. Of course, this 15 bp might actually be fully
explained by the price pressure from trading imbalances as in our model.
In Appendix 6, Panel B, we employ a double-sort methodology in which we independently
sort stocks into one of three Ztj bins (low, medium, and high) and one of three rtj bins (low,
medium, and high) for a total of nine bins. Note that the low Ztj bins consist (on average)
of negative values from selling behavior. Likewise, the low rtj bins consist (on average) of
negative return stocks. Complementary statements can be made about the high bins as well.
The double sort results are shown in Panel B. Within each Ztj category, stocks show positive
momentum. By contrast, sorting stocks by Ztj leads to mean reversion of 36 bp for low
(negative) rtj stocks; 30 bp for medium rtj stocks; and to 45 bp for high (positive) rtj stocks.20
The consistent mean-reversion (after sorting by different return levels) again supports our
model.
6.4
Volume-Based Trading Strategies
Campbell, Grossman, and Wang (1993) show that reversals are prevalent following high
volume periods. The authors suggest the returns of a contrarian portfolio based on highvolume stocks only should be greater than the returns of the contrarian portfolio based on
all stocks (i.e., greater than weekly 15 bp contrarian returns shown in Appendix 6, Panel A).
We compute the ratio of each stock’s weekly turnover to the average of it’s past four week’s
turnover. Each week, we consider only high-volume stocks based on this ratio (stocks in the
top 50%). A contrarian portfolio based on decile sort (as in Panel A) produces an average
20
The average mean-reversion is less than the 72 bp reported in Table 4 because we are no longer using the top
and bottom deciles.
23
weekly return of only 5 bp and not significantly different from zero. We conclude that
the predictive power of our trading imbalances is not subsumed in return and/or volume
measures.
6.5
Transaction Costs
Are returns to the zero-cost portfolio large enough to cover transaction costs? Security firms
in Taiwan are free to charge any commission they like as long as rates do not exceed 0.1425%
of value traded. Large clients typically negotiate rates as low as 0.0700%. A dealer hoping to
trade profitably against uninformed traders should only charge itself marginal cost which we,
being conservative, estimate at 0.0700% of value. In addition, there is a 0.3000% tax levied
on all sales. Thus, we estimate a round-trip transaction cost of 0.4400%. The round-trip
cost on a zero-cost portfolio is 0.8800%. From Figure 2, Panel A we see that the a one week
holding period return is about equal to transactions costs. Holding periods greater than a
week are more than enough to cover transaction costs. A rough calculation based on earning
171 bp (pre-cost) over three weeks translates into over 15% profit per annum after costs.
6.6
Open Prices
We re-form the zero-cost portfolio from our sorting procedure using the next period’s open
price and the sub-sample of 131 stocks. Average weekly returns of the zero-cost portfolio
are 78 bp and are actually higher than 72 bp returns shown in Table 4.
6.7
NTD Shocks
In Section 3 we normalized our empirical trading imbalance measures (Ztj ) to help compare
across stocks. The model is actually specified in dollars (in this case Taiwanese dollars
or “NTD”). We re-estimate Table 3 with weekly, NTD data (and adjust coefficients by a
factor of 106 to make comparisons easier). For Regression 6 we get: π11 =2.8354 with a
8.61 T-statistic; π12 =0.4084 with a 9.32 T-statistic; π21 =-0.8696 with a -4.48 T-statistic;
and π22 =-1565 with a -5.99 T-statistic. The adjusted R2 of the regression is 0.1527 which
is slightly below the value reported in Table 3. Note that the T-statistics are also slightly
lower using Ztj measured in NTD rather than how we define it in Section 3. Overall, though,
the results are similar regardless of which measure we use.
24
6.8
%Ztj Shocks
We test an alternative empirical definition of trading imbalances based on shares held long
on margin.
Shares Held jt
%Ztj ≡
−1
Shares Held jt−1
In many ways, using the measure %Ztj improves the results in this paper. When using %Ztj
instead of Ztj , the zero-cost portfolio earns 45 bp at a daily frequency (T-statistic=18.57)
and 82 bp at a weekly frequency (T-statistic=6.20). When we plot returns of the cumulative
returns to this zero-cost portfolio (as in Figure 1, Panel B) we see complete mean-reversion
back to zero. The zero-cost portfolio has statistically significant profit in each of four equal
sub-periods during our full sample period. We re-estimate Table 3 with weekly data (and
adjust coefficients by a factor of ten). For Regression 6 we get: π11 =1.9310 with a 16.86
T-statistic; π12 =0.0797 with a 13.96 T-statistic; π21 =-0.4064 with a -8.98 T-statistic; and
π22 =-0.0380 with a -9.21 T-statistic. The adjusted R2 of the regression is 0.2284 which is
marginally better than the value reported in Table 3.
6.9
Full Sample
We check that results do not change when we use the full unbalanced panel of 608 stocks. We
re-estimate Table 3, Regression 6 and get: π11 =1.5920 with a 25.68 T-statistic; π12 =0.0499
with a 18.38 T-statistic; π21 =-0.3986 with a -7.90 T-statistic; and π22 =-0.0205 with a -7.35
T-statistic. The adjusted R2 of the regression is 0.2233 which is marginally better than the
value reported in Table 3.
We re-do the weekly sort procedure shown in Table 4 with the full sample of 608 stocks.
The zero-cost portfolio earns an average of 55 basis points per week as opposed to the 72 bp
shown in Table 4. In an expanded sort based on three weeks of past imbalances and tracking
returns for three weeks (as described in Section 6.2), the zero-cost portfolio earns 115 bp.
Thus, results are not qualitatively changed when using the full sample of 608 stocks.
We can also use the full sample to check that results do not change when we use large companies only. We re-estimate Table 3, Regression 6 and include the thirty largest companies
in the market each week (based on market capitalization). The results are: π11 =3.4183 with
a 15.07 T-statistic; π12 =0.4450 with a 8.37 T-statistic; π21 =-0.8291 with a -4.46 T-statistic;
and π22 =-0.1774 with a -4.46 T-statistic. The adjusted R2 of the regression is 0.2059 which is
25
marginally less than the value reported in Table 3. Since we are using 30 companies instead
of 131, we expect T-statistics to be about half of the values reported in Table 3. They are.
Interestingly, the coefficients are almost twice as big. We attribute this finding to the fact
that the large companies have high trading volumes and thus provide a higher signal-to-noise
ratio than the average company in the market.
6.10
Consistent Explanations
The paper uses individual investor trades made through margin accounts. Does this feature drive our results? For example, when prices fall, margin traders face margin calls,
which makes them sell stock. Similarly, when prices go up, margin traders have additional
borrowing capacity and can buy more stock. Such a story is consistent with the positive
contemporaneous correlation of prices and own-stock trading imbalances that we report.
But, such a story cannot explain predictable reversals. If anything, they would suggest price
continuations and not reversals (e.g., selling due to margin calls would push prices down even
further which would lead additional margin calls.) Such a story also fails to provide intuition
for many of our cross-stock findings (e.g., cross-stock impacts are increasing in fundamental
correlation).
Note too that Appendix 3 shows that trading imbalances are not correlated with six month
money market rates. So if borrowing constraints are truly driving margin trading, this
correlation should presumably not be zero. Finally, explanations that margin traders buy in
response to aggregate liquidity shocks are possible. However, for such a story to be consistent
will all of our findings, aggregate liquidity shocks need to induce mean-reversion in prices at
daily, weekly, and monthly frequencies.
There is room in an extended version of our model to put more structure on the trading
imbalances (Ztj ) such as in DeLong et. al. (1990a). But, such structure does not change
an essential feature of our model— Ztj is uncorrelated with (future) fundamentals. Thus,
additional structure might introduce new testable implications but it would not affect the
implications derived and tested in this paper. If we allow strategic behavior by the rational
arbitrageurs—as in DeLong et. al. (1990a)—solutions may quickly become unwieldy. In
short, we believe our model provides a concise and unified framework to explain numerous
relationships between observed trading imbalances and asset prices. Extensions, while the
possible subject of future research, should be able to explain relationships beyond those
addressed by the current model.
26
7
Conclusion
This paper develops a multi-asset model to study how a finite number of risk-averse, nonmyopic investors absorb order imbalances that are uncorrelated with stock fundamentals.
As with earlier single-asset models, our model predicts that prices are contemporaneously
positively correlated with imbalances. Similarly, the model generates predictability of returns even with fully persistent order imbalances. Contributions include the multi-asset
implications the model provides.
We test the implications of our model using an new set of data from the Taiwanese Stock
Exchange. The data are daily and cover hundreds of stocks over an eight-year time period.
The data represent a large pool of trading imbalances coming from a defined set of traders
(individuals who buy stock through margin accounts.) The trading imbalances are publicly
available each trading day. We provide evidence that the traders generating our imbalance
data are unlikely to trade on the basis of private information regarding stock fundamentals.
Such features make our dataset particularly well-suited for testing the empirical implications
of limits-to-arbitrage models.
We show the predictions of price pressure and return predictability are strongly supported
in our data. We achieve high levels of statistical significance and results are robust to several
alternative specifications. We show that an order imbalance to stock k affects stock j when
the two stocks have correlated fundamentals. Once more, empirical results strongly support
the cross-stock effects predicted by the model.
We use a sorting methodology in order to quantify the economic significance of the price
pressure and return predictability. Using a zero-cost portfolio from the sorting, we see that
initial price pressure is 237 bp on an average week. After the formation period, the zero-cost
portfolio exhibits positive expected returns for several weeks. The first week’s return is 72 bp
on average, and cumulative returns reach 188 bp over the five weeks following formation. We
conclude that price pressure and return predictability are economically large and statistically
significant.
Two other novel cross-stock predictions are supported in data. We show that the magnitude
of the contemporaneous cross-stock impact is, on average, four times larger when a pair of
firms belong to the same industry than when the pair come from different industries. We also
show that, within each industry, stocks that are subject to more volatile trading imbalances
tend to have higher variance of returns.
27
Finally, we shed light on the excess co-movement of stock returns in an emerging stock
market. Our model implies that markets with more limited risk bearing capacity (due to
a smaller number of arbitrageurs, for example) display higher excess co-movement of stock
returns with respect to stock fundamentals. Our data indicates that excess co-movement
of returns due to limited risk-bearing capacity is responsible for 54% of the total observed
co-movement in Taiwan.
One theoretical framework (our model) provides a unified explanation for all the empirical
findings in this paper.
28
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33
Figure 1
Cumulative Future Returns of the Zero-Cost Portfolio (Inverted)
This figure shows changes to the price of the (inverted) zero-cost portfolio at the time of formation and during the following ten periods. Z t j is our measure of trading
imbalances over a day or a week. Construction of the measures is described in the text. Shares are sorted at time zero into ten deciles based on Z t j . The long short
portfolio is long shares in the lowest decile of Z t j and short shares in the highest decile. This figure reports results for a sub-sample of 131 firms as described in the text.
The sample period starts 05-Jan-1994 and ends 29-Aug-2002. Data are from the Taiwan Economic Journal.
Panel A: Daily
Panel B: Weekly
2.50%
2.50%
2.00%
2.00%
1.50%
1.50%
0.0042 is next-day’s return
as shown in Table 4
1.00%
1.00%
0.50%
0.50%
0.00%
0.00%
0
1
2
3
4
5
6
7
8
0.0072 is next week’s return
as shown in Table 4
9
0
10
Days after sort
1
2
3
4
5
6
Weeks after sort
34
7
8
9
10
Figure 2
Limits to Arbitrage
This figure shows the cumulative return of the zero-cost portfolio during the ten weeks following formation. Z t j is our measure of trading imbalances over a day or a week.
Construction of the measures is described in the text. Shares are sorted at time zero into ten deciles based on Z t j . The long short portfolio is long shares in the lowest
decile of Z t j and short shares in the highest decile. This figure reports results for a sub-sample of 131 firms as described in the text. The sample period starts 05-Jan-1994
and ends 29-Aug-2002. Data are from the Taiwan Economic Journal.
Panel A: Transaction costs
Panel B: After-Cost Sharpe Ratios
0.030
0.030
SR(0.50)
0.025
0.025
SR(0.40)
0.020
Cumulative portfolio return
Cumulative portfolio return
Our Data
0.015
0.010
0.020
SR(0.30)
0.015
SR(0.20)
0.010
Costs
Costs
0.005
0.005
Our Data
0.000
0.000
1
2
3
4
5
6
7
8
9
1
10
Weeks after sort
2
3
4
5
6
7
Weeks after sort
35
8
9
10
Example
Table 1
Descriptive Statistics of Trading Imbalances
This table gives descriptive statistics of trading imbalances and aggregate holdings in our sample at the individual stock level. Z t j is our measure of trading imbalances
over a day or a week. Holdings at any point in time is the sum of all past trading imbalances: H ti = ∑ Zτj Construction of the measures is described in the text. This
τ = −∞
table also reports the absolute value of trading imbalances. Panel C gives and industry breakdown. The full sample contains 608 firms. The sub-sample contains 131 firms
and is described in the text. The sample period starts 05-Jan-1994 and ends 29-Aug-2002. Data are from the Taiwan Economic Journal.
Panel A: Cross-Sectional Measures of
Z t j (i.e., of Company Time Series Data)
Daily
Full sample
Average
25th-tile
50th-tile
75th-tile
N
Sub-sample
Average
25th-tile
50th-tile
75th-tile
N
Weekly
H tj =
t
∑ Z tj
0.0003
-0.0001
0.0001
0.0004
0.0039
0.0023
0.0037
0.0054
0.0721
0.0361
0.0686
0.1037
608
608
607
607
608
0.0000
0.0000
0.0000
0.0000
0.0012
0.0007
0.0011
0.0016
0.0810
0.0429
0.0782
0.1124
-0.0001
-0.0001
-0.0001
0.0000
0.0036
0.0021
0.0035
0.0048
0.0801
0.0426
0.0773
0.1101
131
131
131
131
131
131
0.0013
0.0007
0.0012
0.0018
608
Z tEW
T
Sub-sample
Average
Stdev.
T
∑Z
τ
0.0725
0.0360
0.0688
0.1049
0.0001
0.0000
0.0000
0.0001
τ = −∞
Daily
Average
Stdev.
t
| Z tj |
| Z tj |
Panel B: Time-Series Measures from an Equally-Weighted Measure (
Full sample
H tj =
Z tj
Z tj
| Z tEW |
= −∞
j
t
Z tEW )
Weekly
H tj =
N
1
N
t
∑ τ∑ Zτ
j
Z tEW
| Z tEW |
j =1 = −∞
H tj =
N
1
N
j =1 = −∞
0.0000
0.0005
0.0014
0.0005
0.0850
0.0216
0.0001
0.0019
0.0042
0.0015
0.0841
0.0220
2359
2359
2360
442
442
443
0.0000
0.0005
0.0012
0.0005
0.0810
0.0214
-0.0001
0.0019
0.0036
0.0015
0.0801
0.0220
2359
2359
2360
442
442
443
36
t
∑ τ∑ Zτ
j
Table 1
Continued
Panel C: Industry Composition
Industry
Full Sample
# of Firms
31-Dec-1998
# of Firms
MktCap (NTD m)
Cement
Foods
Plastics
8
20
20
8
20
18
142,875
182,302
425,803
6
6
13
131,242
106,151
409,139
Textile
Electrical
Wire & Cable
56
33
16
49
24
16
457,651
156,130
169,444
15
5
9
277,598
74,606
147,025
Chemicals
Glass/Ceramics
Pulp/Paper
30
7
7
23
7
6
162,215
60,424
60,137
8
2
3
54,168
45,539
30,289
Steel
Rubber
Automobile
28
9
4
26
9
4
269,150
104,434
172,401
7
6
1
191,370
71,277
59,243
Electronic
Construction
Transportation
221
40
17
115
33
16
3,046,223
271,731
246,019
18
8
6
1,056,146
95,886
90,903
Tourism
Banking
Retailing
Others
7
64
11
10
7
47
11
10
50,303
2,123,875
140,175
152,406
2
9
4
3
20,049
455,953
38,495
67,757
TOTAL
608
449
8,393,698
131
3,422,836
37
Sub-Sample
# of Firms MktCap (NTD m)
Table 2
Unit Root Tests, Cross-Sectional Correlation, and Principal Component Analysis of Trading Imbalances
This table shows trading imbalances are serially uncorrelated but cross-sectionally correlated. Panel A presents the results of an Augmented Dickey-Fuller test for the
presence of a unit root on the level of holdings (cumulative trading imbalances). The null is that the level of holdings follows a random walk. Panel B presents the average
pairwise correlation of contemporaneous trading imbalances. Panel C reports the fraction of contemporaneous trading imbalance correlation explained by the first five
principal components. The sub-sample contains 131 firms and is described in the text. The sample period starts 05-Jan-1994 and ends 29-Aug-2002. Data are from the
Taiwan Economic Journal.
Panel A: Unit Root Tests
We report the number of stocks out of 131 possible, for which the null hypothesis that holdings
follow a random walk is rejected at the 5% and 10% significance levels.
Significance
Levels
Daily
Weekly
5%
3
7
10%
10
16
Panel B: Average Pairwise Correlation of Trading Imbalances
Daily
Weekly
Average
0.0612
0.1209
s.e.
0.0004
0.0001
Fraction of
Pairs with
Corr < 0
3.42%
4.03%
Panel C: Principal Component of Trading Imbalances
(Fraction of Total Correlation Explained)
1st PC
2nd PC
3rd PC
4th PC
5th PC
Daily
Weekly
0.0727
0.0202
0.0178
0.0168
0.0161
0.1347
0.0279
0.0265
0.0208
0.0206
38
Table 3
Price Impacts, Reversals, and Cross-Stock Effects
This table shows both own-stock price impacts and cross-stock price impacts. We present results of a pooled OLS regression of stock j’s return on its own
contemporaneous trading imbalance (coef. π11); the whole market’s (excluding stock j) contemporaneous trading imbalance (coef. π12); stock j’s own lagged trading
imbalance (coef. π21); and the rest of the market’s lagged trading imbalance (coef. π22). “N” is the number of stocks in our sample and equal to 131. The sample period
starts 05-Jan-1994 and ends 29-Aug-2002. Data are from the Taiwan Economic Journal. T-statistics are based on standard errors that allow for heteroscedasticity and
clustering of contemporaneous observations.
j
rt j = π 0 + π 11 Z t j + π 12 (N − 1)Z tEW + π 21 Z t −j 1 + π 22 (N − 1)Z tEW
−1 + ε t
*
*
Daily
π11
π12
π21
Reg. 1
Reg. 2
Reg. 3
Reg. 4
Reg. 5
Reg. 6
Contemp. Own Imbalance
2.7014
---
2.0017
2.6843
---
1.5493
( T-stat )
(44.65)
---
(54.39)
(17.83)
---
(20.33)
Contemp. Cross Imbalance
---
0.1542
---
0.1152
( T-stat )
---
(21.54)
---
(16.08)
-0.4684
-0.7972
-0.4250
-0.4709
(-8.65)
(-26.65)
(-3.13)
(-7.82)
Lagged Own Imbalance
( T-stat )
π22
Weekly
Lagged Cross Imbalance
( T-stat )
Adj. R2
0.0495
0.0015
39
-0.0562
-0.0536
(-7.84)
(-7.88)
0.1730
0.0593
0.0015
0.2128
Table 4
Sort Results
This table shows the results of a sorting procedure based on uninformed trading. In order to compare holdings and changes in holdings across stocks we use Z t j , which is
net shares traded for stock j divided by total shares outstanding. Construction of the measure is described in the text. Each period (day or week) we sort stocks into ten
deciles based on Z t j . The sub-sample contains 131 firms and is described in the text. The sample period starts 05-Jan-1994 and ends 29-Aug-2002. Data are from the
Taiwan Economic Journal. T-statistics are based on standard errors that are robust to heteroscedasticity and autocorrelation.
Daily
Current
Period Decile
( Lowest Z t j ) 1
2
3
4
5
6
7
8
9
j
( Highest Z t t ) 10
Zero-Cost Portfolio
Difference 1-10
T-stat
N
Next
Period
Return
Weekly
Stdev
Next
Period
Return
Stdev
0.0022
0.0008
0.0006
0.0005
0.0001
-0.0001
-0.0003
-0.0005
-0.0008
0.0195
0.0180
0.0172
0.0168
0.0164
0.0161
0.0164
0.0166
0.0177
0.0039
0.0027
0.0018
0.0008
0.0001
0.0001
-0.0002
0.0004
-0.0006
0.0475
0.0439
0.0414
0.0400
0.0408
0.0383
0.0411
0.0432
0.0431
-0.0019
0.0189
-0.0033
0.0462
0.0042
0.0118
0.0072
0.0288
16.77
2,358
5.24
441
40
Table 5
Risk-Adjusted Returns
Returns of our zero-cost portfolio called r1-10 are regressed on the market’s return and other factors’ returns. Construction of the size factor (SMB), the market-to-book
factor (HML), and the momentum factor (MOM) are all described in the text. Results are based on the sub-sample of 131 firms that is described in the text. The sample
period starts 05-Jan-1994 and ends 29-Aug-2002. Data are from the Taiwan Economic Journal.
rt1−10 = α + β MKT (rt m − rt f ) + β SMB rt SMB + β HML rt HML + β MOM rt MOM + ε t
Reg. 1
Reg. 2
Reg. 3
Reg. 4
Reg. 5
Constant ( α )
0.0073
0.0074
0.0073
0.0080
0.0079
( T-stat )
(5.27)
(5.39)
(5.20)
(6.14)
(6.04)
Market ( βMKT )
0.0499
---
---
---
0.0185
( T-stat )
(1.32)
---
---
---
(0.49)
SMB ( βSMB )
-0.0191
---
---
0.0028
( T-stat )
(-0.82)
---
---
(0.10)
HML ( βHML )
-0.0094
---
0.0026
( T-stat )
(-0.52)
---
(0.14)
MOM ( βMOM )
-0.1346
-0.1340
( T-stat )
(-6.54)
(-6.60)
Std. errs
White
White
White
White
White
N
441
441
441
441
441
41
Table 6
The Magnitude of Cross-Stock Price Impact
This table shows the relationship between two stocks fundamental correlation and the cross-stock price impact. For each pair of stocks “j” and “k” we estimate the
regression below. The contemporaneous cross-stock coefficient is labeled CCC j , k and is the average of stock k’s effect on stock j and stock j’s effect on stock k. In the
equations, Z t j represents stock j’s trading imbalances; Z tk represents stock k’s trading imbalances; and ( N − 2) Z tEW is the sum across all trading imbalances except
stock j and stock k. The sub-sample contains the 131 firms for which we have a balanced panel and is described in the text. The sample period starts 05-Jan-1994 and ends
29-Aug-2002. Data are from the Taiwan Economic Journal.
**
rt j = π 0j + π 111 Z t j + π 112 Z tk + π 113 ( N − 2)Z tEW + π 121 Z t j−1 + π 122 Z tk−1 + π 123 ( N − 2)Z tEW
+ ε tj
−1
**
**
rt k = π 0k + π 211 Z t j + π 212 Z tk + π 213 ( N − 2)Z tEW + π 221 Z t j−1 + π 222 Z tk−1 + π 223 ( N − 2) Z tEW
+ ε tk
−1
**
CCC j , k ≡
**
π 112 + π 211
2
All Pairs
Pairs With j and k in
Same Industry
Pairs With j and k in
Different Industries
Average CCC j,k
0.2851
0.9973
0.2342
Median CCC j,k
0.2024
0.7661
0.1715
Stdev CCC j,k
0.7669
1.0792
0.7130
Num. Pairs
8,515
567
7,948
42
Table 7
Excess Co-Movement
This table reports the average R2 from regressions of individual stock returns on factors. We consider each of the 131 firms for which we have a balanced panel. Individual
stock returns are labeled rt j . Factors include: the market excess return ( rt m − rt f ); the portion of the market excess return that is orthogonal to a market-wide measure of
trading imbalances ( rt m ⊥ Z
EW
); the portion of the market-wide measure of trading imbalances that is parallel to stock j’s trading imbalance ( Z tEW // j ); and the portion of
the market-wide measure of trading imbalances that is orthogonal to stock j’s trading imbalance ( Z tEW ⊥ j ). The sample period starts 05-Jan-1994 and ends 29-Aug-2002.
Data are from the Taiwan Economic Journal.
Specification
Regression Used to Generate R2 Measures
Average R2
Stdev. of R2
Num. Stocks
0.2958
0.0945
131
0.1630
0.0905
131
1
rt j − rt f = α + β m (rt m − rt f ) + ε t j
2
rt j − rt f = γ 0 + γ 1 rt m⊥ Z
3
rt j − rt f = γ 0 + γ 2 (Z tEW // j ) + ν t j
0.0827
0.0624
131
4
rt j − rt f = γ 0 + γ 3 (Z tEW ⊥ j ) + ν t j
0.1062
0.0396
131
5
rt j − rt f = γ 0 + γ 1 Z tm⊥ Z
0.3474
0.0911
131
(
(
EW
EW
)+ν
j
t
) + γ (Z
2
EW // j
t
) + γ (Z
3
EW ⊥ j
t
43
) +ν
j
t
Appendix 1: Model
At date 3 assets pay liquidating dividends which we can write as P3j = (S1j + S2j + Φj3 ) and
P3k = (S1k + S2k + Φk3 ). At date 2, we solve for prices and demands. Risk averse agents choose
holdings in order to maximize date 3 wealth. The CARA-normal framework implies the
following optimal demand at date 2:
#
#
"
#!−1 "
"
X2j
S1j + S2j + Φj3 − P2j
S1j + S2j + Φj3
1
=
E2
V ar2
λ
X2k
S1k + S2k + Φk3 − P2k
S1k + S2k + Φk3
"
X2j
X2k
#
1
=
λ
"
VΦ ρφ VΦ
ρφ VΦ VΦ
#−1 "
S1j + S2j − P2j
S1k + S2k − P2k
#
Market clearing at date 2 implies: X2j = 1 − (Z1j + Z2j ) and X2k = 1 − (Z1k + Z2k ). Substituting
the market clearing condition into the optimal demand equation and solving for prices yields
prices at date 2.
On date 1, risk averse agents choose holdings in order to maximize date 3 wealth, given their
optimal decisions at date 2 and their knowledge about future equilibrium prices. The date 3
wealth is:
W3 = X2j (P3j − P2j ) + X2k (P3k − P2k ) + X1j (P2j − P1j ) + X1k (P2k − P1k )
We substitute the optimal X2j , X2k and the equilibrium values of P3j , P3k , P2j , P2k into the
equation
j k directly above to get the wealth equation as a function of two control variables
X1 , X1 :
W3
= X1j S1j + λVΦ Z1j + ρφ Z1k − RVΦ − P1j + X1k S1k + λVΦ Z1k + ρφ Z1j − RVΦ − P1k
−Z1j Φj3 + X1j S2j + λVΦ (X1j + 2Z1j − 1) + ρφ X1k + 2Z1k − 1
Z2j − Z1k Φk3 + X1k S2k + λVΦ ((X1k + 2Z1k − 1) + ρφ X1j + 2Z1j − 1 )Z2k
−Z2j Φj3 + λVΦ (Z2j )2 + 2ρλVΦ Z2j Z2k − Z2k Φk3 + λVΦ (Z2k )2
The risk averse investors must choose X1j , X1k in order to maximize E[e−λW3 ], given an
observed realization of S1j , S1k . It turns out that W3 is a quadratic function of normal
random variables. Therefore, we can apply the following result:
A
Let Y ∼ N (µ, Σ) and A is symmetric, then:
E[exp(C + B 0 Y − Y 0 AY )] =
− 1
1
−1 |Σ| 2 2A + Σ−1 2 exp(C + B 0 µ + µ0 Aµ + (B 0 − 2µ0 A0 )(2A + Σ−1 )−1 (B − 2Aµ))
2
Where Y ≡
X1j
1 − Z1j
h
Φj3
S2j
X1k =
Z2j
i
Φk3 S2k Z2k . The market clearing
1 − Z1k . Setting derivatives with respect to
conditions at date 1 are
=
and
X1j and X1k equal to zero,
and using the market clearing condition at date 1, yields the equilibrium prices at date 1.
B
Appendix 2: Sub-sample Selection
Not all stocks in Taiwan are actively held on margin. Some stocks have stock price information when they enter our sample, but do not have any shares held on margin. At a later time,
stocks begin to be held on margin. There is a “build-up period” as holdings go from zero
shares to some steady-state level. During the build-up period, the level of margin holdings
increases in a predictable fashion until it reaches a steady state. Other stocks in our sample
see margin holdings disappear during the sample period. There is a “wind-down period”
in which the level of margin holdings decreases in a predictable fashion. Finally, there are
stocks where the number of shares held on margin does not change very often.
The graph below shows the level of margin holdings for two individual stocks. The first
stock is ticker 1215. This stock has non-zero margin holdings throughout our sample period.
The second stock in the graph below is ticker 2328. This stock has no margin holdings until
June 1994. At that time, margin holdings build up rapidly until they reach a similar level
to other stocks. There is wind-down period before January 1996, followed by a very quick
rise to more normal levels. For most of 1999, margin holdings decline until they hit zero.
Between June 2000 and June 2001, there are no holdings data for ticker 2328. After June
2001, margin holdings again build up until they reach levels that are commensurate with
the rest of the market. In early 2002 we cease to have margins holding data for ticker 2328
although price data continue.
The TSE has two situations that lead to a predictable trend in the level of margin holdings.
The first is when there is a capital change by a firm; the second is when a firm reports poor
operating performance. Either situation can cause the TSE to declare that all long margin
positions must be closed out within a pre-determined time frame. It is around such episodes
that margin holdings act predictably. For the reasons discussed above and shown in the
graph below, we run many of our tests on a sub-sample of companies with positive margin
holdings for the entire time the stock is listed. There are 131 such companies. We carry out
all tests with the full sample of 608 firms as well as the sub-sample of 131 firms.
C
Appendix 2 (continued)
Holdings for Two Individual Stocks
This figure shows the fraction of two companies’ shares from our sample. In order to compare holdings we use H τj = ∑ Zτj which is defined as shares held long on margin
for stock j divided by total shares outstanding. Construction of this measures is described in the text. The first company (ticker=1215) has positive holdings for the entire
time it is listed. The second stock (ticker=2328) has a one-year period where holdings in our sample disappear completely from the data (the stock is no longer held on
margin). This period is from 1999 to 2000. Thus, ticker 1215 is included in our balanced panel (sub-sample of 131 companies) and stock 2328 is not. The sample period
starts 05-Jan-1994 and ends 29-Aug-2002. Data are from the Taiwan Economic Journal.
1215
0.20
2328
0.15
0.10
0.05
D
Jul-02
Jan-02
Jul-01
Jan-01
Jul-00
Jan-00
Jul-99
Jan-99
Jul-98
Jan-98
Jul-97
Jan-97
Jul-96
Jan-96
Jul-95
Jan-95
Jul-94
0.00
Jan-94
Fraction of shares held long on margin
0.25
Appendix 3: Additional Tests Relating to Ztj
The volatility of tracking error (difference in returns between the mimicking portfolio and
the market) is 1.106% per week. Such a large value indicates that investors who generate
our data are not (in aggregate) using their margin accounts to track the market. In other
words, they are not using margin accounts to lever-up the market portfolio. The mimicking
portfolio has a market beta that is close to, but statistically more than one.
Trading is also very noisy. The absolute value of the portfolio changes by 0.40% per week on
average. But, the number of shares held per stock experiences an average absolute change
of 5.04% per week. All of these overview statistics lead us to believe our sample of data
represents a significant pool of uninformed trades.
There is further evidence that investors do not use margin accounts to hold a levered position
in the market. Holding levels do not move together as can be seen by the the fact the the
average pair-wise correlations in Table 2, Panel B are significantly less than one.
Correlation Table
We show our net trading imbalance measure (Ztj ) has low correlation with economic variables.
Series
Freq.
Corr with
ZtEW
P-Value
Num. of
Obs.
Change in 6 mo money mkt rate
Change in Premium on Taiwan Fund
Change in Comm. Paper Rate
Loans at Domestic Banks
Daily
Weekly
Monthly
Monthly
<0.01
-0.07
-0.07
-0.04
0.95
0.13
0.50
0.71
2,256
438
104
104
E
Appendix 4
Expanded Sort Results
This table shows expanded results of a sorting procedure based on uninformed trading. In order to compare holdings and changes in holdings across stocks we use Z t j ,
which is net shares traded for stock j divided by total shares outstanding. Construction of the measure is described in the text. We sort stocks into ten deciles based on the
past 1, 2, or 3 three values of Z t j at both daily and weekly frequencies. We form a portfolio that goes long the bottom decile and short the top decile of the sorted stocks.
We then track that portfolio for the next 1, 2, or 3 periods (days or weeks). The sub-sample contains 131 firms and is described in the text. The sample period starts 05-Jan1994 and ends 29-Aug-2002. Data are from the Taiwan Economic Journal. T-statistics are based on standard errors that are robust to heteroscedasticity and
autocorrelation.
Return over Future Days
# of Formation
Days in Past
1
2
1
0.0042
(16.77)
2
3
Return over Future Weeks
3
# of Formation
Weeks in Past
1
2
3
0.0055
0.0064
1
0.0072
0.0104
0.0130
(14.18)
(13.29)
(5.24)
(5.13)
(4.92)
0.0037
0.0051
0.0061
0.0066
0.0104
0.0142
(14.84)
(12.12)
(11.04)
(4.40)
(4.69)
(4.79)
0.0033
0.0048
0.0056
0.0067
0.0117
0.0171
(13.24)
(11.13)
(9.51)
(4.82)
(5.18)
(5.45)
2
3
F
Appendix 5: Factor Construction
We construct our own Fama-French size (SMB) and book-to-market (HML) portfolios as
well as a Carhart momentum (MOM) portfolio. We do this at both daily and weekly frequencies. After construction, we compare our daily time series to similar time series that
were graciously provided by Brad Barber, Neil Yi-Tsung Lee, Yu-Jane Liu, and Anlin Chen.
The correlation between our series and theirs is high for SMB and HML. Our MOM series is
not highly correlated with Chen’s due to the fact that he sorts on the past one-year returns
and we sort on past days or week’s returns (as we explain below). The test results presented
in this paper are at a weekly frequency.
Size portfolio
We construct a zero-cost size portfolio, also known as “small minus big” or “SMB.” Each
week we sort stocks into deciles based on market capitalization (as of the Friday close). SMB
is the return on the portfolio that goes long the bottom 30% of stocks (the smallest three
deciles)and short the top 30% of stocks (the largest or biggest three deciles). Portfolios are
equally weighted and held for one week.
Market-to-book portfolio
We construct a zero-cost market-to-book portfolio, also known as “high minus low” or
“HML.” Each week we sort stocks into deciles based on their book-to-market ratios (the
market prices are as of the Friday close and the book values are as of the previous December). HML is the return on the portfolio that goes long the top 30% of stocks (the high
or top three deciles) and short the bottom 30% of stocks (the low or bottom three deciles.)
Portfolios are equally weighted and held for one week.
Momentum portfolio
We construct a zero-cost momentum portfolio, known also as “MOM.” Each week we sort
stocks into deciles based on the previous week’s return (the Friday close to Friday close).
MOM is the return on the portfolio that goes long the top 30% of stocks (the winners) and
short the bottom 30% of stocks (the losers). Portfolios are equally weighted and held for one
week.
G
Appendix 6
Return Momentum
This table tests for momentum and volume effects. Panel A shows the results of a traditional (single sort) test for momentum. Panel B sorts independently by current
returns ( rt j ) and current trading imbalances ( Z t j ). All results show next period’s returns ( rt +j 1 ). The sub-sample contains 131 firms and is described in the text. The
sample period starts 05-Jan-1994 and ends 29-Aug-2002. Data are from the Taiwan Economic Journal.
Panel A: Single-Sort on Returns (Momentum) Results
Next Period’s Returns ( rt +j 1 ) Shown
Daily
Weekly
Next
Period
Return
Stdev
8
9
( Highest rt j t ) 10
-0.0016
-0.0001
0.0001
-0.0002
-0.0002
-0.0002
-0.0003
-0.0000
0.0001
0.0027
0.0198
0.0182
0.0175
0.0170
0.0167
0.0165
0.0164
0.0171
0.0181
0.0200
0.0017
0.0019
0.0006
0.0010
0.0009
0.0009
-0.0004
-0.0015
-0.0006
0.0002
0.0507
0.0428
0.0414
0.0416
0.0407
0.0407
0.0405
0.0405
0.0422
0.0495
Momentum Port. 10-1
0.0042
0.0171
-0.0015
0.0419
Contrarian Port. 1-10
-0.0042
0.0171
0.0015
0.0419
Current
Period Decile
( Lowest rt j ) 1
2
3
4
5
6
T-stat
N
-6.71
2,360
Next
Period
Return
0.76
442
H
Stdev
Appendix 6
Continued
Panel B: Independent Double-Sort by Trading Imbalances and Returns
Next Week’s Returns ( rt +j 1 ) Shown
Low Z t j
Medium Z t j
High Z t j
Effect From Z t j
Low ri,t
0.0017
0.0001
-0.0019
-0.0036
Medium ri,t
0.0015
0.0008
-0.0015
-0.0030
High ri,t
0.0034
0.0008
-0.0011
-0.0045
Effect From ri,t
0.0017
0.0007
0.0008
I